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Chapter 10   Geometry

10-4    Finding Volumes

 

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Volume of a Rectangular Solid

A rectangular solid is a 3-dimensional version of a plane rectangle.

 

 

Equation

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 The equation for the volume of any rectangular solid is:

V = lwh

Where:

  • l is the length
  • h is the height
  • w is the width
  • V is the volume of the figure

Example

Determine the volume of a rectangular solid that is 10 inches long, 15 inches wide, and 8 inches high.

Problem   Find the volume of a rectangular solid where l = 10 in, w = 15 in, and h = 8 in.

Cite the appropriate equation

 V = lwh  

Assign the given values

 V = 10 · 15 · 8  

Complete the solution
V = 1200  
Solution   V = 1200 cubic in
  or
V = 1200 in3

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

 

Example

 

  

Thinking Mathematically

 

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

Volume of a Cube

 

Equation

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 The equation for the volume of any cube:

V = s3

Where:

  • s is the length of the sides
  • V is the volume of the cube

Example

One side of a cube measures 10 in. What is the volume of this cube?

Problem   Find the volume of a cube that measures 10 in to a side.

Cite the appropriate equation

 V = s3  

Assign the given values

 V = 103  

Expand the expression
V = 1000  
Solution   V = 1000 cubic in
  or
V = 1000 in3

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

 

Example

 

  

Thinking Mathematically

A cube is actually a special kind of rectangular solid. For a rectangular solid, the only the parallel sides are equal to one another -- and there are three groups of parallel sides. For a cube, all sides are equal. Therefore the equation for the volume of a rectangualr solid:

V = lwh

becomes

V = s · s · s

or

V = s3

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

Volume of a Pyramid

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A pyramid

 

Equation

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 The equation for the volume of a pyramid:

V = 1/3b2h

Where:

  • b is the length of the bases
  • h is the height
  • V is the volume of the pyramid

 

Example

Determine the volume of a pyramid that has a base of 2 m and a height of 3 m.

Problem   Find the volume of a cone where b = 2 m and h = 3 m.

Cite the appropriate equation

 V = 1/3b2h  

Assign the given values

 V = 1/3 · 22 · 3  

Complete the solution
V = 4  
Solution   V = 4 m3

The volume is 4 cubic meters.

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

 

Example

 

  

Thinking Mathematically

 

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

Volume of a Cylinder

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A cylinder

 

Equation

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 The equation for the volume of a cylinder:

V = pr2h

Where:

  • r is the radius
  • h is the height of the cylinder
  • V is the volume of the cylinder

Example

A certain cylinder has a radius of 2.5 inches and a height of 8 inches. What is the volume of this cylinder?

Problem   Find the volume of a cylinder where r = 2.5 in and h = 8 in.

Cite the appropriate equation

 V = pr2h  

Assign the given values

 V = 3.14 · 2.52 · 8  

Complete the solution
V = 157  
Solution   V = 157 in3

The volume is s157 cubic inches

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

 

Example

 

  

Thinking Mathematically

 

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

 

 

 

 

Volume of a Sphere

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A sphere

 

 

Equation

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 The equation for the volume of a sphere is:

V = 4/3pr3

Where:

  • p = approximately 3.14 or 22/7
  • r is the radius of the sphere
  • V is the volume of the sphere

 

Example

Determine the volume of a sphere that has a radius of 10 ft.

Problem   Find the volume of a sphere where r = 10 ft.

Cite the appropriate equation

 V = 4/3pr3  

Assign the given values

 V = 4/3 · 3.14 · 103  

Complete the solution
V = 4186.667

Rounding to the nearest 1s unit:
     V = 4187

  
Solution   V = 4187 ft3

The volume is slightly less than 4187 cubic feet

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

 

Example

 

  

Thinking Mathematically

 

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

 

Volume of a Cone

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A cone

 

 

 

Equation

fig100407.gif (1752 bytes)

 The equation for the volume of a cone:

V = 1/3pr2h

Where:

  • p = approximately 3.14 or 22/7
  • r is the radius of the cone
  • h is the height of the cone
  • V is the volume of the cone

 

Example

Determine the volume of a cone that is 12 cm high, and has a radius of 2 cm.

Problem   Find the volume of a cone where r = 2 cm and h = 12 cm.

Cite the appropriate equation

 V = 1/3pr2h  

Assign the given values

 V = 1/3 · 3.14 · 22 · 12  

Complete the solution
V = 50.24

Rounding to the nearest 1s unit:
     V = 50

  
Solution   V = 50 cm3

The volume is slightly over 50 cubic centimeters.

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

 

Example

 

  

Thinking Mathematically

 

 

Exercises

Click the  ?  symbol to see the correct answer.

1.    =  ?  2.   =  ?  3.    =  ?  4.    =  ?  5.    =  ? 
6.    =  ?  7.    =  ?  8.    =  ?  9.    =  ?  10.  =  ? 

 

 

Author: David L. Heiserman
Publisher: SweetHaven Publishing Services

Copyright © 2006, David L. Heiserman
All Rights Reserved